Morris was born in Crouch End, North London, and grew up in Muswell Hill. He is the son of South African-born Glen and Zimbabwean-born Andrea Morris, and spent a lot of time in both countries growing up. He has a younger sister, Amy. His parents are both from Jewish families from Eastern Europe. He resides in Los Angeles. Morris is a close friend of Pretty Little Liars co-star Ian Harding.

Morris began acting at the Anna Scher Theatre in London. It was there, when he was 13, that a director on the hit British drama The Knock spotted him and asked him to play a boy pushed into the drug trade. Morris continued his intensive training under Anna Scher for the next eight years. In this period more opportunities arose and after appearing in several television shows, films and commercials; he went on to spend three seasons with the Royal Shakespeare Company. Working with directors such as Steven Pimlott, Tim Albery and Elijah Moshinsky, and alongside actors like Simon Russell Beale, Samuel West and Roger Allam. Morris is currently working in the United States, having played the lead in the unsold NBC pilot Young Arthur (2002) and appearing in the film Cry Wolf (2005) as Owen Matthews and in the film Whirlygirl (2006) as James Edwards.

Morris was next seen in the independent horror film Donkey Punch as Josh, critically honoured after its screening at the 2008 Sundance Film Festival. He also appeared in the Tom Cruise-starring Valkyrie and the Sugababes video "Freak Like Me". Morris next had a supporting role in the slasher film Sorority Row as Andy. In 2010, he played the recurring role of Agent Owen in 24.

Morris was part of the cast of the 2010 ABC documentary-style dramedy series My Generation, playing "The Rich Kid" Anders Holt. In July 2012, Morris joined the ABC series Once Upon a Time as Prince Phillip.

Morris starred as Victor in the film Kelly + Victor, which premiered on 16 October 2012 at the BFI London Film Festival, and was released theatrically in the UK on 20 September 2013. The film won the BAFTA Award for Outstanding Debut by a British Writer, Director or Producer in 2014. Morris has finished filming the lead in Universal's Dragonheart 3: The Sorcerer's Curse.

In March 2014, Morris was cast as Reverend Paul Curtis in the Amazon Studios series Hand of God. In August 2014, he landed a recurring guest role on the Fox sitcom New Girl, playing Ryan Geauxinue.